# diffeotopy

Let $M$ be a manifold and $I=[0,1]$ the closed unit interval. A smooth map $h\colon M\times I\rightarrow M$ is called a diffeotopy (on $M$) if for every $t\in I$:

 $h_{t}:=h(-,t)\colon M\rightarrow M$

is a diffeomorphism.

Two diffeomorphisms $f,g\colon M\to M$ are said to be diffeotopic if there is a diffeotopy $h\colon M\times I\to M$ such that

1. 1.

$h_{0}=f$, and

2. 2.

$h_{1}=g$.

Remark. Diffeotopy is an equivalence relation among diffeomorphisms. In particular, those diffeomorphisms that are diffeotopic to the identity map form a group.

Two points $a,b\in M$ are said to be isotopic if there is a diffeotopy $h$ on $M$ such that

1. 1.

$h_{0}=id_{M}$, the identity map on $M$, and

2. 2.

$h_{1}(a)=b$.

Remark. If $M$ is a connected manifold, then every pair of points on $M$ are isotopic.

Pairs of isotopic points in a manifold can be generazlied to pairs of isotopic sets. Two arbitrary sets $A,B\subseteq M$ are said to be isotopic if there is a diffeotopy $h$ on $M$ such that

1. 1.

$h_{0}=id_{M}$, and

2. 2.

$h_{1}(A)=B$.

Remark. One special example of isotopic sets is the isotopy of curves. In $\mathbb{R}^{3}$, curves that are isotopic to the unit circle are the trivial knots.

Title diffeotopy Diffeotopy 2013-03-22 14:52:43 2013-03-22 14:52:43 rspuzio (6075) rspuzio (6075) 9 rspuzio (6075) Definition msc 57R50 isotopic diffeotopic